A fourth-place finish would have given the Spartans (21-10, 8-8) a game with the fifth-place finisher in the Big Ten on Friday, which meant a bye on the first day of Big Ten Tournament play.
The Spartans have won the last eight contests against the Wildcats, including a 66-45 victory in East Lansing earlier this season.
Despite being under the weather, Neitzel led MSU with 22 points against the Badgers, including hitting a clutch jumper that put the Spartans ahead with under a minute to play.
Goran Suton also stepped up and scored 16 points for the Spartans, who had Morgan, their second-leading scorer, in foul trouble throughout the game.
MSU has a record of 10-7 in the tournament's nine-year history, and it will only be the second time that they will play on the opening day of the tourney.
After reeling off four-consecutive home wins down the stretch, Michigan State has now lost their last two games, although both on the road.
Now they must contain the fire and regroup as they try to build their NCAA Tournament resume with last-minute impressions and wins along the way.
The Spartans won the tournament in 1999 and 2000, and lost in the quarterfinals in 1998, 2001, 2002 and 2005. MSU lost in the semifinals in 2003, 2004 and 2006.
March Madness is coming towards full swing as championship week arrives in college basketball.